I connected the Powerbook to an external display. Even though it's not registering on the external display, it caused the Powerbook's screen to be bright again. I've opened Disk Utility from the Install CD and am running that now. Strange!

Just to review:

Sometimes the screen goes black (without computer turning off) seconds after it's started up normally with the chime and apple logo.

Sometimes it doesn't start up at all but just emits a deflated little spin-down sound when I've pressed the "on" button.

Usually I can't shut it down except by unplugging it and removing the battery.

Recently, I was able to get it into Target Disk Mode for hours and copy things from it.

Just now I'm able to start up from an install CD (something it wouldn't do before), though then the screen went 90% dark. I could barely see information on the screen. When I hooked it to an external monitor, the external monitor is just all blue, but the Powerbook screen is now regularly illuminated. So now I'm running Disk Utility.

I swear I'm not making this up! This is the craziest thing I've ever tried to troubleshoot. Clearly I was in the total rookie leagues when I troubleshot things before.